DALLAS — The questions being raised by right-wing bloggers and conservative commentators about a Pentagon training exercise called Jade Helm 15 traverse the outer edges of political paranoia.

The eight-week exercise starting in July and planned for locations in Texas, New Mexico, California and other Southwestern states, they say, is part of a secret plan to impose martial law, take away people’s guns, arrest political undesirables, launch an Obama-led hostile takeover of red-state Texas, or do some combination thereof.

But when Gov. Greg Abbott issued a directive to the Texas State Guard to keep watch over the military operation, seeming to give those concerns some official credence, it raised a more credible question: To what degree has the extreme become mainstream in Texas?

In his directive, issued last week to Maj. Gen. Gerald Betty, commander of the Guard, Mr. Abbott wrote that he wanted Jade Helm 15 monitored because it was “important that Texans know their safety, constitutional rights, private property rights and civil liberties will not be infringed.”

Mr. Abbott has defended his decision to monitor the exercise, denying that he was legitimizing conspiracy theories and explaining that he was responding to, and gathering information for, concerned Texans. “There was, frankly, an overreaction to the simple fact that someone has to be in charge with gathering and disseminating information,” he told reporters on Monday. “And we stepped in to play that role, which is a role to be applauded.”

But even some Republicans say Mr. Abbott’s reaction, and the failure of any Republicans in office to clearly denounce what many see as extremism and paranoia, speak volumes about how far right the party has moved in a state in which it controls all the levers of power.

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A meeting in Bastrop County, Tex., to discuss plans for the Jade Helm 15 military exercise drew a large crowd, including a number of skeptics.CreditJay Janner/Austin American-Statesman, via Associated Press

“I have been an eyewitness to the sort of radicalization of Republican primary politics in Texas, but this was just a shocking extension of this fanaticism,” Todd A. Smith, a Republican lawyer and former state legislator, said in an interview. In a letter to the governor’s office, he criticized Mr. Abbott for “pandering to idiots.”

In the interview, Mr. Smith added, “I was really shocked to read that a governor was issuing press releases employing the Texas State Guard at taxpayer expense to protect us from the U.S. military, and that he felt like that was either something that was important from a policy perspective or necessary from a political perspective. Either one is unbelievable to me.”

The Army’s Special Operations Command at Fort Bragg, N.C., announced the exercise in March, saying that while a multistate training operation was not unique, “the size and scope of Jade Helm sets this one apart.” Jade Helm 15, according to military officials, amounts to a giant war game spread across the Southwest, with Army Green Berets and other elite personnel sharpening their skills in terrain they might find overseas.

Army officials said the operation would not be conducted at Fort Hood, the sprawling base in Killeen, Tex., but instead would take place in remote areas on “private and public land with the permission of the private landowners, and from state and local authorities.” They said the most noticeable effect on local communities will be “an increase in vehicle and military air traffic and its associated noise.”

Mr. Abbott’s directive has touched off a wave of condemnation and support, with some blasting him for giving conspiracy-minded activists an air of respectability and others saying that they, too, were concerned about federal military drills on Texas soil.

Residents in rural Bastrop County packed a community meeting last week about the plans for the military exercise. On Tuesday, Representative Louie Gohmert, a Republican, called for changes to the exercise, particularly a war-simulation map that labels Texas, Utah and part of Southern California as “hostile.” He said that because the Obama administration believed “that major threats to the country include those who support the Constitution, are military veterans or even ‘cling to guns or religion,’ patriotic Americans have reason to be concerned.”

A leader of the Republic of Texas, a separatist group that believes Texas never officially joined the United States, forwarded an email to reporters in which an anonymous man claiming to be a Texas Ranger wrote that Texans “should be pushing back against JH15.”

Chuck Norris has also weighed in, questioning, in a commentary on the conservative website WND, “those who are pulling the strings at the top of Jade Helm 15 back in Washington.” Walmart has responded as well, in the other direction — it dismissed rumors that tunnels were being built by the military beneath closed stores, including one in Midland, Tex., as part of a Pentagon-led takeover.

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Gov. Greg Abbott issued a directive to the Texas State Guard to watch the military operation to ensure that Texans’ rights were not infringed.CreditDeborah Cannon/American-Statesman, via Associated Press

On CNN on Tuesday, David Dewhurst, a Republican and former lieutenant governor, kept to the fine line that some other Texas Republican figures have — praising Mr. Abbott, reiterating the state’s deep support of the military and suggesting that he understood the concerns of Texans about Jade Helm 15.

“I think you’ve got some paranoia, which is based upon legitimate concerns by my fellow Texans, by myself and a lot of Americans about the trustworthiness and the competence of President Obama, but let’s not project that on to our military,” Mr. Dewhurst told the network.

Senator Ted Cruz told Bloomberg News in an interview that he understood the concerns about the operation given a federal government that for six years has been “disrespecting the liberty of its citizens.”

On the other hand, the Texas Democratic Party called on Mr. Abbott to apologize to military personnel. Critics say that in Texas, where Republicans control both chambers of the Legislature and every statewide elective office, far-right activists have the power to sway Republican primary elections, giving them disproportionate influence.

Representative Joaquín Castro, a Democrat, said some fellow members of Congress had told him they were appalled the governor would order a state body to watch the United States military. “It’s dangerously irresponsible for a governor to fan the flames of conspiracy and paranoia against our own military and government,” Mr. Castro said.

In recent years, numerous Republican elected officials in Texas have publicly embraced the ideology, conspiracy theories and anti-Obama sentiments of the far right. Some warned of civil unrest after Mr. Obama’s re-election and suggested some support for the Texas secession movement, even meeting with its leaders in the Texas Capitol.

In Bastrop County outside Austin, the top elected official, County Judge Paul Pape, said military leaders in January made a presentation to county officials about Jade Helm 15 and were given permission to conduct the training exercise in the area. After news of the exercise circulated on social media, county leaders asked military officials to return for a public meeting.

About 200 people turned out for the April 27 meeting. “The people who came with an open mind I think got some useful information and I hope left with a higher comfort level,” Mr. Pape said. “The ones that came with their minds made up probably left with their minds made up.”

David Montgomery contributed reporting from Austin, Tex.

A version of this article appears in print on , Section A, Page 15 of the New York edition with the headline: Military Exercise Stirs Conspiracy Theorists. Order Reprints | Today’s Paper | Subscribe